Lesson 2: Evaluating Trig Functions at Special Angles
Estimated time: 35-40 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- State the exact values of sine, cosine, and tangent at π/6, π/4, and π/3
- Use reference angles to extend these values to all four quadrants
- Evaluate all six trig functions at special angles
- Identify the complete set of unit circle coordinates for multiples of 30° and 45°
The Special Right Triangles
Two special right triangles produce the exact trig values that form the backbone of the unit circle:
45-45-90 triangle: sides in ratio 1 : 1 : √2. On the unit circle (hypotenuse = 1), this gives coordinates (√2/2, √2/2) at 45°.
30-60-90 triangle: sides in ratio 1 : √3 : 2. On the unit circle, this gives coordinates (√3/2, 1/2) at 30° and (1/2, √3/2) at 60°.
First Quadrant Special Values
| θ | Degrees | cos θ | sin θ | tan θ |
|---|---|---|---|---|
| 0 | 0° | 1 | 0 | 0 |
| π/6 | 30° | √3/2 | 1/2 | √3/3 |
| π/4 | 45° | √2/2 | √2/2 | 1 |
| π/3 | 60° | 1/2 | √3/2 | √3 |
| π/2 | 90° | 0 | 1 | undefined |
Memory tip: For sine at 0°, 30°, 45°, 60°, 90°, think √0/2, √1/2, √2/2, √3/2, √4/2 which gives 0, 1/2, √2/2, √3/2, 1. Cosine is the same list in reverse order.
Reference Angles
Reference Angle — The acute angle formed between the terminal side of θ and the x-axis. It is always between 0° and 90° (0 and π/2).
To find the reference angle θ′:
- QI: θ′ = θ
- QII: θ′ = π − θ (or 180° − θ)
- QIII: θ′ = θ − π (or θ − 180°)
- QIV: θ′ = 2π − θ (or 360° − θ)
Example 1: Finding a Reference Angle
Find the reference angle for 5π/6.
Solution: 5π/6 is in QII. θ′ = π − 5π/6 = π/6. The reference angle is π/6 (30°).
Example 2: Evaluating Using Reference Angles
Find sin(5π/4).
Step 1: 5π/4 is in QIII (between π and 3π/2). Reference angle: 5π/4 − π = π/4.
Step 2: sin(π/4) = √2/2.
Step 3: In QIII, sine is negative. So sin(5π/4) = −√2/2.
Example 3: Evaluating cos(11π/6)
Step 1: 11π/6 is in QIV. Reference angle: 2π − 11π/6 = π/6.
Step 2: cos(π/6) = √3/2.
Step 3: In QIV, cosine is positive. So cos(11π/6) = √3/2.
Evaluating Tangent at Special Angles
Since tan θ = sin θ / cos θ, we can find tangent from the sine and cosine values.
Example 4: Finding tan(2π/3)
Step 1: 2π/3 is in QII. Reference angle: π − 2π/3 = π/3.
Step 2: sin(π/3) = √3/2, cos(π/3) = 1/2.
Step 3: In QII: sin is +, cos is −. So sin(2π/3) = √3/2, cos(2π/3) = −1/2.
Step 4: tan(2π/3) = (√3/2)/(−1/2) = −√3.
The Complete Unit Circle
By applying reference angles and sign rules to the three key first-quadrant angles, we can fill in all 16 standard positions on the unit circle (multiples of 30° and 45°). This is one of the most important things to memorize in trigonometry.
Strategy for memorizing: Learn the first-quadrant values, then use symmetry. Each quadrant mirrors QI with appropriate sign changes.
Check Your Understanding
1. Find sin(3π/4).
2. Find cos(4π/3).
3. Find tan(7π/6).
Key Takeaways
- The 45-45-90 triangle gives √2/2 for both sin and cos at 45°.
- The 30-60-90 triangle gives the values at 30° and 60°.
- Reference angles let you extend QI values to all four quadrants.
- Apply the correct sign based on the quadrant (ASTC).
- tan θ = sin θ / cos θ, so tangent can be computed from the other two.
Ready for More?
Next Lesson
Lesson 3 covers right triangle definitions of trig functions (SOH-CAH-TOA).
Start Lesson 3